Question

What is Pythagoras theorem??

Explain in detail with diagram....

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Answer 1

Pythagoras theorem statement:-

Pythagoras theorem states that "In a right -angled triangle ,In a square of the hypotenuse side is equal to the sum of square of the other two sides "

• The side of this triangle have been named as perpendicular ,Base and hypotenuse .
• here ,the hypotenuse is the longest side ,as it is opposite to the angle 90°.
• The sides of a right triangle which have positive integers values ,when squared are put in equation ,also called a Pythagoras triple

History:-

The Pythagoras is named after a Greek mathematician called Pythagoras .

Pythagoras Theorem Formula :-

(AB)² +(BC)²=(AC)² (According to figure )

Provation:-

$\bold { ƓƖƔЄƝ:-}$

• ∆ABC in which LABC is 90°

$\bold{To \:prove :-}$

• (AB)² +(BC)²=(AC)²

$\bold{Construction:-}$

• BD perpendicular AC

$\bold{Proof:-}$

• In ∆ABC and ∆ ADB,we have

LA=LA

LABC=LADB. (90°each)

Hence ,by $\huge\mathfrak{\green{❝AA❞}}$ similarity criterion ∆ABC~∆ADB.

So we can say that

$\frac{AD}{AB}$=$\frac{AB}{AC}$

=> AB² =AD×AC -----equation (1)

• In ∆ CBA and ∆ CDB

LC = LC (common angle)

LCBA=LCDB (90°each)

hence ,by $\huge\mathfrak{\purple{❝AA❞}}$similarity criterion ∆CBA~∆CDB

Hence we can say that

$\frac{BC}{DC}$=$\frac{AC}{BC}$

=>(BC)² =AC×DC ------ equation (2)

From equation (1) and (2)

adding equation (1) and (2)

• (AB)²+(BC)² =AC×DC + AD×AC

• (AB)²+(BC)² =AC(DC+ AD)

• (AB)²+(BC)² =AC(AC)

• $\huge\star\underline\mathrm\pink{(AB)² +(BC)²=(AC)² }$

$\huge\color{red}\boxed{\colorbox{black}{Hence\:proved}}$

Application of Pythagoras theorem:-

• To know if triangle is right angled or not
• To find diagonal of a square
• In a right angled triangle we can calculate the length of any side if the two other are given

How to use ?

For example if the value of AB=3 and BC=4 and AC =?

we know ,(AB)² +(BC)²=(AC)²

put value

(3)² +(4)²=(AC)²

9+16=(AC)²

25=(AC)²

√25=AC

5=AC

Hence ,the third side is 5 cm

As we can see AB+BC >AC

3+4 >5

7 > 5

hence ,5cm is hypotenuse of the given triangle

Answer 2

${\bigstar \: {\large{\pmb{\sf{\underbrace{\underline{RequirEd \; solution \: is \: mentioned}}}}}}}$

Phythagoras Theorm: Phythagoras Theorm is applied only in right angled triangles. Phythagoras Theorm states that in a right angled triangle, the square of the hypotenuse is always equal to the sum of the square of base and the sum of the square of perpendicular of the right angled triangle. Perpendicular is also termed as height

A triangle is right angled triangle if the square of the hypotenuse is equal to the sum of the square of base and the sum of the square of perpendicular.

• Kindly see the attachment 1st

Right angled triangle is always equal to 90° and the other two triangle's are acute.

The side that is opposite to the right angle is known as hypotenuse.

The other two sides are known as base or perpendicular/height or legs of the right angled triangle.

Hypotenuse is the longest side of a right angled triangle.

Formula: Formula that is used to calculate pythagoras theorem is mentioned below. Let us see what is it!

${\small{\underline{\boxed{\red{\pmb{\sf{(Hypotenuse)^{2} \: = (Base)^{2} + \: (Height)^{2}}}}}}}}$

By this formula we can derive more formulas regards phythagoras theorm. Let us see what are they!

${\small{\underline{\boxed{\pmb{\sf{\star \: (Hypotenuse)^{2} \: - (Height)^{2} \: = (Base)^{2}}}}}}}$

${\small{\underline{\boxed{\pmb{\sf{\star \: (Hypotenuse)^{2} \: - (Base)^{2} \: = (Height)^{2}}}}}}}$

Converse of Phythagoras Theorm:

In a triangle, if the square of the length of the longest side of the triangle is equal to the sum of the squares of the length of the other two sides, the triangle is said to be a right angled triangle. This property is called the converse of phythagoras theorm.

Now let us verify the property of phythagoras theorm by taking an example:

In a triangle△ABC, AB is equal to 12 centimetres, BC is equal to 9 centimetres and AC is equal to 15 centimetres

• (Hint: BC is the base, AC is hypotenuse and AB is the height)

${\small{\underline{\boxed{\pmb{\sf{(Hypotenuse)^{2} \: = (Base)^{2} + \: (Height)^{2}}}}}}} \\ \\ :\implies {\pmb{\sf{(Hypotenuse)^{2} \: = (Base)^{2} + \: (Height)^{2}}}} \\ \\ :\implies {\pmb{\sf{(15)^{2} \: = (9)^{2} + (12)^{2}}}} \\ \\ :\implies {\pmb{\sf{225 \: = 81 + 144}}} \\ \\ :\implies {\pmb{\sf{225 \: = 225}}} \\ \\ {\pmb{\sf{L.H.S = R.HS}}} \\ \\ {\pmb{\sf{Henceforth, \: verified!}}}$

Now let us verify pythagoras theorem:

Firstly we have to draw three same right angled triangle. (You can take hint from attachment 1st) Note: Make same right angled triangles. Give then same names. Afterthat let us name sides as a,b and c then what the squares be? a²,b² and c². Now let us do the sum of two sides. What we get? Hmm? Nice! Correct we get equal, remember what phythagoras theorm says!? Henceforth, verified.